Products Problem with 3 variables

Question

Hi need some help with the following Question please

I write down three positive numbers a, b and c.

The product of a and b is 2. The product of b and c is 24. The product of c and a is 3.

What is the sum of all three numbers?

Answer 1

$$ab=2\implies b=\frac{2}{a}$$ $$bc=24\implies b=\frac{24}{c}$$ $$\frac2a=\frac{24}c\implies c=12a$$ $$ac=a(12a)=12a^2=3\implies a=\frac{1}{2}$$ $$c=12\left(\frac12\right)=6$$ $$b=\frac2{1/2}=4$$ $$\therefore a+b+c=\frac12+4+6=\frac{21}{2}$$

Answer 2

You solve the system $ab=2,~bc=24,~ ac=3$ in order to get $a,b,c$. Using the first equation you get for example $a=\frac{2}{b}$ and put into other two equations, and so on. You get $a=\frac{1}{2},~b=4,~c=6$. Then you can compute the sum easily as $10.5$.